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PROTEIN PHYSICS LECTURE 8

PROTEIN PHYSICS LECTURE 8. THERMODYNAMISC & STATISTICAL PHYSICS. WHAT IS “TEMPERATURE”? EXPERIMENTAL DEFINITION :. EXPERIMENTAL DEFINITION. = t, o C + 273.16 o. WHAT IS “TEMPERATURE”?. THEORY Closed system: energy E = const. S ~ ln[M].

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PROTEIN PHYSICS LECTURE 8

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  1. PROTEIN PHYSICS LECTURE 8

  2. THERMODYNAMISC & STATISTICAL PHYSICS

  3. WHAT IS “TEMPERATURE”? EXPERIMENTAL DEFINITION : EXPERIMENTAL DEFINITION = t,oC + 273.16o

  4. WHAT IS “TEMPERATURE”? THEORY Closed system: energy E = const S ~ln[M] CONSIDER: 1 state of “small part” with  & all states of thermostat with E-. M(E-) = 1•Mth(E-) St(E-) = k •ln[Mt(E-)]  St(E) - •(dSt/dE)|E Mt(E-) = exp[St(E)/k] • exp[-•(dSt/dE)|E/k]

  5. All-system’s states with E have equal probabilities For “small part’s” state: depends on e: COMPARE: Probability1(1) = Mt(E-1)/SE’M(E’) ~ exp[St(E)/k] • exp[-1• (dSt/dE)|E/k] and Probability1(1) ~ exp(-1/kBT) (BOLTZMANN) One has: (dSt/dE)|E = 1/T k = kB ______________________________________________________________   -kBT, M  Mexp(1)  M2.72

  6. (dSth/dE) = 1/T P1(1) ~ exp(-1/kBT) Pj(j) = exp(-j/kBT)/Z(T); j Pj(j)  1 Z(T) = i exp(-i/kBT) partition function СТАТИСТИЧЕСКАЯ СУММА

  7. Along tangent: S-S(E1)= (E-E1)/T1 i.e.,F = E - T1S= const (= F1 = E1 - T1S1) stable Unstable (explodes, V → ): Unstable (fells):  unstable 

  8. Separation of potential energy in classic (non-quantum) mechanics: P() ~ exp(-/kBT) Classic:=COORD+ KIN KIN= mv2/2 : does not depend on coordinates Potential energy COORD: depends only on coordinates P() ~ exp(-COORD/kBT) • exp(-KIN/kBT) Z(T) = ZCOORD(T)•ZKIN(T)  F(T) = FCOORD(T)+FKIN(T) ======================================================================================================================== Elementary volume: (mv)x = ħ  (x)3 =(ħ/|mv|)3

  9. IN THERMAL EQUILIBRIUM: TCOORD=TKIN=T We may consider further only potential energy: EECOORD MMCOORD S(E)SCOORD(ECOORD ) F(E)FCOORD , etc.

  10. TRANSITIONS: THERMODYNAMICS

  11. gradual transition “all-or-none” (or first order) phase transition coexistence (T/T*)(E/kT*)~1 coexistence & jump

  12. “all-or-none” (or first order) phase transition F(T1)

  13. Second order phase transition change Not observed in proteins up to now: they are too small

  14. TRANSITIONS: KINETICS

  15. n#=nexp(-F#/kBT) n# n  TRANSITION TIME: t01 = t0#1 = = # (n/n#)=#exp(+F#/kBT)

  16. PARALLEL REACTIONS: TRANSITION RATE = SUM OF RATES (or: the highest rate) RATE = 1/TIME

  17. # # _ start finish CONSECUTIVE REACTIONS: TRANSITION TIME  SUM OF TIMES t0… finish = t0#1 finish + t0#2 finish + …

  18. #main #main _ _ “trap”: on “trap”: out start start finish finish TRANSITION TIME IS ESSENTIALLY EQUAL FOR “TRAPS” AT AND OUT OF PATHWAYS OF CONSECUTIVE REACTIONS: TRANSITION TIME  SUM OF TIMES (or: the longest time)

  19. DIFFUSION: KINETICS

  20. Mean kinetic energy of a particle:mv2/2 ~ kBT<> = j Pj(j) j v2 = (vX2)+(vY2)+(vZ2)Maxwell:

  21. Friction stops a molecule within picoseconds: m(dv/dt) = -(3D)v [Stokes law] D – diameter; m ~ D3 – mass;  – viscosity tkinet 10-13 sec  (D/nm)2in water During tkinet the molecule moves somewhere by Lkinet ~ v•tkinet Then it restores its kinetic energy mv2/2 ~ kBT from thermal kicks of other molecules, and moves in another random side CHARACTERISTIC DIFFUSION TIME: nanoseconds

  22. Friction stops a molecule within picoseconds: tkinet 10-13 sec  (D/nm)2in water During tkinet the molecule moves somewhere by Lkinet ~v•tkinet Then it restores its kinetic energy mv2/2 ~ kBT from thermal kicks of other molecules, and moves in another random side CHARACTERISTIC DIFFUSION TIME: nanoseconds The random walk allows the molecule to diffuse at distance D (~ its diameter) within ~(D/L kinet)2steps, i.e., within tdifft tkinet•(D/Lkinet)2 4•10-10 sec  (D/nm)3in water

  23. For “small part”: Pj(j) = exp(-j/kBT)/Z(T); Z(T) = j exp(-j/kBT) j Pj(j) = 1 E(T) = <> = j j Pj(j) if allj =: #STATES = 1/P, i.e.: S(T) = kBln(1/P) S(T) = kB<ln(#STATES)>= kBj ln[1/Pj(j)]Pj(j) F(T) = E(T) - TS(T) = -kBT  ln[Z(T)] STATISTICAL MECHANICS

  24. Thermostat: Tth = dEth/dSth “Small part”: Pj(j,Tth) ~ exp(-j/kBTth); E(Tth) = j jPj(j,Tth) S(Tth) = kBj ln[1/Pj(j,Tth)]Pj(j,Tth) Tsmall_part = dE(Tth)/dS(Tth) = Tth STATISTICAL MECHANICS

  25. Along tangent: S-S(E1)= (E-E1)/T1 i.e., F = E - T1S= const (= F1 = E1 - T1S1)

  26. Separation of potential energy in classic (non-quantum) mechanics: P() ~ exp(-/kBT) Classic:=COORD+ KIN KIN= mv2/2 : does not depend on coordinates Potential energy COORD: depends only on coordinates P() ~ exp(-COORD/kBT) • exp(-KIN/kBT) ZKIN(T) =K exp(-K/kBT): don’t depend on coord. ZCOORD(T) =Cexp(-C/kBT): depends on coord. Z(T) = ZCOORD(T)•ZKIN(T)  F(T) = FCOORD(T)+FKIN(T) ======================================================================================================================== Elementary volume: (mv)x = ħ  (x)3 =(ħ/|mv|)3

  27. P(KIN+COORD) ~ exp(-COORD/kBT)•exp(-KIN/kBT) P(COORD) = exp(-COORD/kBT) / ZCOORD(T) ZCOORD(T) =Cexp(-C/kBT): depends ONLY on coordinates P(KIN) = exp(-KIN/kBT) / ZKIN(T) ZKIN(T) =K exp(-K/kBT): don’t depend on coord. T<0: unstable (explodes) <KIN>   at T<0 due to P(KIN) ~ exp(-KIN/kBT)

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